Name Class Time Math 2250 Extra Credit Problems Chapter 10 S2015 Submitted work. Please submit one stapled package with this sheet on top. Kindly check-mark the problems submitted and label the paper Extra Credit . Label each solved problem with its corresponding problem number, e.g., Xc10.3-20 . Problem Xc10.3-20. (Inverse transform) Solve for f (t) in the relation L(f (t)) = s4 1 . Use partial fractions in the details. − 8s2 + 16 Problem Xc10.3-24. (Inverse transform) Solve for f (t) in the relation L(f (t)) = s4 1 s , showing the details that give the answer f (t) = 2 sinh at sin at 4 + 4a 2a Problem Xc10.4-12. (Inverse transform, convolution) 1 . Instead of the convolution theorem, use partial fractions for the s(s2 + 4s + 5) details. If you can see how, then check the answer with the convolution theorem. Solve for f (t)) in the relation L(f (t)) = Problem Xc10.4-26. (Inverse transform techniques) Use the s-differentiation theorem in the details of solving for f (t) in the relation L(f (t)) = arctan to apply the theorem lims→∞ L(f (t)) = 0. 3 . You will need s+2 Problem Xc10.4-40. (Series methods for transforms) √ cos 2 t . Then find L(f (t)) as a series by Laplace Expand in a series, using Taylor series formulas, the function f (t) = √ πt transform of each √ series term, separately. Finally, re-constitute the series in variable s into elementary functions, namely e−1/s divided by s. Problem Xc10.5-6. (Second shifting theorem, Heaviside step) Find the function f (t) in the relation L(f (t)) = se−s . s2 + π 2 Problem Xc10.5-14. (Transforms of piecewise functions) Let f (t) = cos πt 0 ≤ t ≤ 2, Find L(f (t)). Details should expand f (t) as a linear combination of Heaviside step 0 t > 2. functions. Problem Xc10.5-26. (Sawtooth wave) Let f (t + a) = f (t) and f (t) = t on 0 ≤ t ≤ a. Then f is a-periodic and has a Laplace transform obtained from the 1 e−as periodic function formula. Show the details in the derivation to obtain the answer L(f (t)) = 2 − . as s(1 − e−as ) Problem Xc10.5-28. (Modified sawtooth wave) Let f (t + 2a) = f (t) and f (t) = t on 0 ≤ t ≤ a, f (t) = 0 on a < t ≤ 2a. Then f is 2a-periodic and has a Laplace transform obtained from the periodic function formula. Derive a formula for L(f (t)). The answer to this problem can be found in Edwards-Penney, section 10.5. Problem Xc-EPbvp-7.6-8. (Impulsive DE) Solve by Laplace methods x00 + 2x0 + x = δ(t) − 2δ(t − 1), x(0) = 1, x0 (0) = 1. Check the answer in maple using dsolve({de,ic},x(t),method=laplace) . Problem Xc-EPbvp-7.6-18. (Switching circuit) A passive LC-circuit has battery 6 volts and model equation i00 + 100i = 6δ(t) − 6δ(t − 1), i(0) = 1, i0 (0) = 1. The switch is closed at time t = 0 and opened again at t = 1. Solve the equation by Laplace methods and report the number of full cycles observed before the steady-state i = 0 is reached (to two decimal places). Check the answer in maple using dsolve({de,ic},i(t),method=laplace) . End of extra credit problems chapter 10. 2